We consider a class of probability measures $\mu_{s,r}^{\alpha}$ which have
explicit Cauchy-Stieltjes transforms. This class includes a symmetric beta
distribution, a free Poisson law and some beta distributions as special cases.
Also, we identify $\mu_{s,2}^{\alpha}$ as a free compound Poisson law with
L\'{e}vy measure a monotone $\alpha$-stable law. This implies the free infinite
divisibility of $\mu_{s,2}^{\alpha}$. Moreover, when symmetric or positive,
$\mu_{s,2}^{\alpha}$ has a representation as the free multiplication of a free
Poisson law and a monotone $\alpha$-stable law. We also investigate the free
infinite divisibility of $\mu_{s,r}^{\alpha}$ for $r\neq2$. Special cases
include the beta distributions $B(1-\frac{1}{r},1+\frac{1}{r})$ which are
freely infinitely divisible if and only if $1\leq r\leq2$.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ473 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm